Properties

Label 1-161-161.18-r0-0-0
Degree $1$
Conductor $161$
Sign $0.288 - 0.957i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.888 + 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (0.723 − 0.690i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.327 + 0.945i)10-s + (−0.888 − 0.458i)11-s + (0.0475 + 0.998i)12-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.327 − 0.945i)16-s + (−0.995 + 0.0950i)17-s + (0.235 + 0.971i)18-s + (−0.995 − 0.0950i)19-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (0.723 − 0.690i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.327 + 0.945i)10-s + (−0.888 − 0.458i)11-s + (0.0475 + 0.998i)12-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.327 − 0.945i)16-s + (−0.995 + 0.0950i)17-s + (0.235 + 0.971i)18-s + (−0.995 − 0.0950i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.288 - 0.957i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.288 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3207480499 - 0.2382882923i\)
\(L(\frac12)\) \(\approx\) \(0.3207480499 - 0.2382882923i\)
\(L(1)\) \(\approx\) \(0.5147685667 + 0.01051064367i\)
\(L(1)\) \(\approx\) \(0.5147685667 + 0.01051064367i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (-0.786 + 0.618i)T \)
5 \( 1 + (0.723 - 0.690i)T \)
11 \( 1 + (-0.888 - 0.458i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (0.235 - 0.971i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.981 + 0.189i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.580 - 0.814i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.261059664800715483786770277406, −27.05026750797330766052804103240, −26.13086102849265798807707818933, −25.266754488908002786347689794642, −24.257076022429706681686472033521, −23.052101257613064349145291054436, −21.905946306640401655517589903403, −21.31518413765008564843137714642, −19.857196453528836052109109640259, −18.843982009380141570426098698000, −18.12152886716273742238095423983, −17.40830009483715455625268495780, −16.52736701399994769438011859135, −15.19459072086243954758764382276, −13.577189153098399573223538278869, −12.667687672023245103463808434747, −11.55282397028699599920351351443, −10.6065990632481513770334564273, −9.86022825261593362486951446189, −8.36542992317315708764070705833, −7.03444110968362443273969694853, −6.47207025497748746165876497367, −4.78682156559699504615590721498, −2.64128554417084100303508779662, −1.768909133334418120514779325, 0.46224107513211102642004846020, 2.34701838723272150903133055776, 4.67190213807171097509250170356, 5.60616386871273128568799665667, 6.503533442555402723489204237644, 8.105644757027087462560216789249, 9.15893493533499040333848155552, 10.179486032885956830730374152579, 10.846084125856544273779161447303, 12.24560281888346297413838684963, 13.51667946080898540092770456247, 15.11638817341249535073077208163, 15.80381498130677960368884657357, 16.91275812563289189518086401107, 17.438945981683085465223844429996, 18.320490384408124519370819449965, 19.71171893410325666545858345169, 20.79220322739210080689824011256, 21.55490515829585022201365853184, 22.8582519706043769379451558826, 23.96511762337168668966030984737, 24.66434258744990602615957498424, 25.83315894879847080981646173822, 26.73087531596626934988776212916, 27.5798088961218005623749221209

Graph of the $Z$-function along the critical line